Adm No: 2222291¶

Name: Toh Kien Yu¶

Background Research¶

I will be forecasting Gas, Electricity and Water Consumption for this project.

1. Forecasting gas consumption is essential for setting up and running of a sustainable system.

2. Forecasting electricity is important is crucial as it serves as a foundation to make decisions in power sector planning.

3. Forecasting water consumption is also important to ensure people do not face service interruptions are able to maintain an affordable rate while being dependable.

Overall, forecasting allows us to better plan and make decisions when managing our resource.

Importing Necessary Libraries¶

In [1]:
import numpy as np
import pandas as pd
import sklearn
import matplotlib.pyplot as plt
import seaborn as sns
import warnings
warnings.filterwarnings('ignore')
sns.set()

Data Loading¶

Date is set to index¶

In [2]:
df = pd.read_csv('dataset/Energy Consumption Dataset.csv')
df['DATE'] = pd.to_datetime(df['DATE'],format='%d/%m/%Y')
df.set_index('DATE', inplace=True)
df.head()
Out[2]:
Gas Consumption (tons) Electricity Consumption (MWh) Water Consumption (tons)
DATE
1990-01-01 18.0 725.1 548.8
1990-02-01 15.8 706.7 640.7
1990-03-01 17.3 624.5 511.1
1990-04-01 18.9 574.7 515.3
1990-05-01 22.0 553.2 488.4

Data Exploration¶

In [3]:
print('Dataset Shape')
print(df.shape)

Dataset Shape
(397, 3)
In [4]:
df.describe()
Out[4]:
Gas Consumption (tons) Electricity Consumption (MWh) Water Consumption (tons)
count 397.000000 397.000000 397.000000
mean 23.785139 888.472544 484.953652
std 4.903452 153.877594 133.908863
min 11.600000 553.200000 44.400000
25% 20.200000 771.100000 384.400000
50% 23.500000 897.800000 487.400000
75% 27.900000 1005.200000 580.200000
max 46.000000 1294.000000 811.000000
In [5]:
df.info()

<class 'pandas.core.frame.DataFrame'>
DatetimeIndex: 397 entries, 1990-01-01 to 2023-01-01
Data columns (total 3 columns):
 #   Column                         Non-Null Count  Dtype  
---  ------                         --------------  -----  
 0   Gas Consumption (tons)         397 non-null    float64
 1   Electricity Consumption (MWh)  397 non-null    float64
 2   Water Consumption (tons)       397 non-null    float64
dtypes: float64(3)
memory usage: 12.4 KB

No Null Values¶

In [6]:
# No Null Values
df.isnull().sum()
Out[6]:
Gas Consumption (tons)           0
Electricity Consumption (MWh)    0
Water Consumption (tons)         0
dtype: int64

Data Visulization¶

In [10]:
dfWithoutYr2023 = df[df.index.year != 2023]
yearlyGasMean = dfWithoutYr2023.resample('Y')['Gas Consumption (tons)'].mean()
yearlyElectricityMean = dfWithoutYr2023.resample('Y')['Electricity Consumption (MWh)'].mean()
yearlyWaterMean = dfWithoutYr2023.resample('Y')['Water Consumption (tons)'].mean()

Gas Consumption (tons) Trend¶

In [11]:
plt.figure(figsize = (10,4))
plt.plot(yearlyGasMean)
plt.title('Gas Consumption (tons) Over The Years')
plt.xlabel('Year')
plt.ylabel('Gas Consumption (tons)')
plt.show()

There is an increase of about 7 tons of gas consumption over the years with fluctuations occurring during this period.¶

Electricity Consumption (MWh) Trend¶

In [12]:
plt.figure(figsize = (10,4))
plt.plot(yearlyElectricityMean)
plt.title('Electricity Consumption (MWh) Over The Years')
plt.xlabel('Year')
plt.ylabel('Electricity Consumption (MWh)')
plt.show()

Electricity Consumption has been increasing over the years which can be possibly attributed to the global population increase.¶

Water Consumption (tons) Trend¶

In [13]:
plt.figure(figsize = (10,4))
plt.plot(yearlyWaterMean)
plt.title('Water Consumption (tons) Over The Years')
plt.xlabel('Year')
plt.ylabel('Water Consumption (tons)')
plt.show()

While there is a general fall in water consumption, fluctuations occurred greatly where water consumption peaked at 675 tons in 2010.¶

Gas, Electricity and Water Consumption's Monthly Mean Over The Years¶

In [14]:
# Monthly mean of gas/electricity/water consumption
peakGas = df.groupby(df.index.month)[['Gas Consumption (tons)']].mean()
peakElectricity = df.groupby(df.index.month)[['Electricity Consumption (MWh)']].mean()
peakWater = df.groupby(df.index.month)[['Water Consumption (tons)']].mean()

fig = plt.figure(figsize=(10,8))
fig.set_facecolor('lightblue')
ax = fig.add_subplot(311)
plt.suptitle('Gas, Electricity and Water Consumption Monthly Mean over Year 1990-2023')
plt.plot(peakGas)
plt.title('Gas Consumption (tons)')
plt.xlabel('Month')
plt.ylabel('Gas Consumption (tons)')

ax2 = fig.add_subplot(312)
plt.plot(peakElectricity)
plt.title('Electricity Consumption (MWh)')
plt.xlabel('Month')
plt.ylabel('Electricity Consumption (MWh)')

ax3 = fig.add_subplot(313)
plt.plot(peakWater)
plt.title('Water Consumption (tons)')
plt.xlabel('Month')
plt.ylabel('Water Consumption (tons)')
plt.tight_layout()
plt.show()

From the graph above,

1. Gas Consumption and Electricity Consumption tend to peak in January

2. Water Consumption is highest in June

Gas, Electricity and Water consumption in recent years 2021 and 2022¶

In [15]:
#Gas,Electricity and Water consumption in recent years 2021 and 2022
year2021 = df[(df.index.year == 2021)]
year2022 = df[(df.index.year == 2022)]

fig = plt.figure(figsize=(11,10))
fig.set_facecolor('lightblue')
ax = fig.add_subplot(311)
plt.suptitle('Gas, Electricity and Water Consumption in Year 2021-2022')
plt.plot(year2021.index.month,year2021['Gas Consumption (tons)'],label='2021')
plt.plot(year2022.index.month,year2022['Gas Consumption (tons)'],label='2022')
plt.title('Gas Consumption (tons)')
plt.xlabel('Month')
plt.ylabel('Gas Consumption (tons)')
plt.legend()
ax2 = fig.add_subplot(312)
plt.plot(year2021.index.month,year2021['Electricity Consumption (MWh)'],label='2021')
plt.plot(year2022.index.month,year2022['Electricity Consumption (MWh)'],label='2022')
plt.title('Electricity Consumption (MWh)')
plt.xlabel('Month')
plt.ylabel('Electricity Consumption (MWh)')
plt.legend()
ax3 = fig.add_subplot(313)
plt.plot(year2021.index.month,year2021['Water Consumption (tons)'],label='2021')
plt.plot(year2022.index.month,year2022['Water Consumption (tons)'],label='2022')
plt.title('Water Consumption (tons)')
plt.xlabel('Month')
plt.ylabel('Water Consumption (tons)')
plt.legend()
plt.tight_layout()
plt.show()

From the graph above, it provides data in recent years 2021 and 2022. Electricity and Water Consumption usually peaks in August.¶

In [84]:
sns.pairplot(df)
Out[84]:
<seaborn.axisgrid.PairGrid at 0x22f2aa38d90>
In [85]:
df.hist()
plt.show()
In [86]:
sns.heatmap(df.corr(),annot=True)
Out[86]:
<Axes: >

Check Stationarity On Time Series¶

Time Series are stationary if they do not have a trend or seasonal effects. A stationary series's mean, variance and covariance do not vary with time.

To check for stationarity of the time series, we will do a Visual Check and Augmented Dickey-Fuller Test

1. Visual Check¶

In [16]:
# Visual Check
fig, ax = plt.subplots(3, 1, figsize=(9, 7))
df['Gas Consumption (tons)'].plot(ax=ax[0])
df['Electricity Consumption (MWh)'].plot(ax=ax[1])
df['Water Consumption (tons)'].plot(ax=ax[2])
ax[0].set_title('Gas Consumption (tons)')
ax[1].set_title('Electricity Consumption (MWh)')
ax[2].set_title('Water Consumption (tons)')
plt.tight_layout()
plt.show()

From the visual check,

Gas consumption and Water Consumption are a stationary time series, as we do not see a trend in the series, and covariance is a function of time.

For Electricity Consumption, the time series is non-stationary as the mean varies with tim, and there is an upward trend.

2. Augmented Dickey-Fuller Test¶

Augmented Dickey-Fullter Test is a popular statistic test and is useful for determining whether the time series is stationary.¶

H$_0$ (Null Hypothesis): The time series can be represented by a unit root, that is not stationary.

H$_1$ (Alternative Hypothesis): The time series is stationary

In [17]:
# Augmented Dickey-Fuller Test
from statsmodels.tsa.stattools import adfuller
result1 = adfuller(df['Gas Consumption (tons)'])
result2 = adfuller(df['Electricity Consumption (MWh)'])
result3 = adfuller(df['Water Consumption (tons)'])
print('Gas Consumption (tons) p-value: %f' % result1[1])
print('Electricity Consumption (MWh) p-value: %f' % result2[1])
print('Water Consumption (tons) p-value: %f' % result3[1])

Gas Consumption (tons) p-value: 0.010811
Electricity Consumption (MWh) p-value: 0.186218
Water Consumption (tons) p-value: 0.000090

Gas Consumption (tons) p value is 0.010811 which is < 0.05. Reject the null hypothesis (H0), the data does not have unit root and is stationary

Electricity Consumption (MWh) p value is 0.186218 which is > 0.05. Fail the reject the null hypothesis (H0), the data has a unit root and is non-stationary.

Water Consumption (tons) p value is 0.000090 which is < 0.05. Reject the null hypothesis (H0), the data does not have unit root and is stationary.

Electricity Consumption is differenced in order to make the time series stationary¶

In [18]:
from statsmodels.tsa.statespace.tools import diff
from statsmodels.tsa.stattools import adfuller
df_diff = diff(df['Electricity Consumption (MWh)'], k_diff=1)
result1 = adfuller(df['Electricity Consumption (MWh)'])
result2 = adfuller(df_diff)
print('Electricity Consumption (MWh) before differencing p-value: %f' % result1[1])
print('Electricity Consumption (MWh) after differencing p-value: %f' % result2[1])

fig, ax = plt.subplots(1, 2, figsize=(14, 4))
df['Electricity Consumption (MWh)'].plot(ax=ax[0])
df_diff.plot(ax=ax[1])
ax[0].set_title('Time Series Before Differencing')
ax[1].set_title('Time Series After Differencing')
plt.show()

Electricity Consumption (MWh) before differencing p-value: 0.186218
Electricity Consumption (MWh) after differencing p-value: 0.000000

Seasonal Decomposition¶

In [19]:
from statsmodels.tsa.seasonal import seasonal_decompose
result = seasonal_decompose(df['Gas Consumption (tons)'][330:], model='additive')
result.plot()
plt.show()

Seasonal period observed every 12 months.¶

In [20]:
from statsmodels.tsa.seasonal import seasonal_decompose
result = seasonal_decompose(df['Electricity Consumption (MWh)'], model='additive')
result.plot()
plt.show()

Seasonal period observed every 12 months.¶

In [21]:
from statsmodels.tsa.seasonal import seasonal_decompose
result = seasonal_decompose(df['Water Consumption (tons)'], model='additive')
result.plot()
plt.show()

Seasonal period observed every 12 months.¶

Modelling

We will forecast Gas Consumption first, Water Consumption then Electricity Consumption.

Evaluation Metrics: Mean Absolute Percentage Error (Percentage difference between predicted and actual values)

Defining Train And Test Data¶

An 90:10 ratio between training and test data is chosen where data before Year 2019-11 is used for training and data from Year 2019-11 is used for testing¶

In [22]:
hDF = df.copy()
train_data = hDF[hDF.index<'2019-11']
test_data = hDF[hDF.index>='2019-11']
(len(train_data)/(len(test_data) + len(train_data))) * 100
Out[22]:
90.17632241813602
In [23]:
gasTrain = train_data[['Gas Consumption (tons)']]
waterTrain = train_data[['Water Consumption (tons)']]
electricityTrain = train_data[['Electricity Consumption (MWh)']]
In [24]:
gasTest = test_data[['Gas Consumption (tons)']]
waterTest = test_data[['Water Consumption (tons)']]
electricityTest = test_data[['Electricity Consumption (MWh)']]

Gas Consumption¶

Exponential Smoothing (As a Baseline)¶

In [25]:
from statsmodels.tsa.holtwinters import ExponentialSmoothing
from sklearn.metrics import mean_absolute_percentage_error
expoModel = ExponentialSmoothing(gasTrain,trend='add',seasonal='add',seasonal_periods=12).fit()
pred = expoModel.forecast(len(gasTest))
mape = mean_absolute_percentage_error(gasTest,pred)*100

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
In [26]:
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(gasTrain.index,gasTrain, label='training')
ax.plot(gasTest.index,gasTest, label='testing')
ax.plot(gasTest.index, pred, label='prediction')
plt.legend()
plt.title('Exponential Smoothing')
plt.show()
In [27]:
print(f'Test Data Mean Absolute Percentage Error: {mape:.2f}%')

Test Data Mean Absolute Percentage Error: 23.13%

ACF and PACF Plot¶

In [28]:
from statsmodels.graphics.tsaplots import plot_acf
from statsmodels.graphics.tsaplots import plot_pacf
plot_acf(hDF['Gas Consumption (tons)'],lags=50)
plot_pacf(hDF['Gas Consumption (tons)'],lags=30)
plt.show()

Tail off at ACF Plot, which indicates an AR model. From PACF plot, cut off happens at lag 2. This illustrates an AR(2) model¶

Gas Consumption Auto Arima Model¶

Auto Arima seeks to find the most ideal parameter to help us get started.¶

In [29]:
from pmdarima.arima import auto_arima
arima_model = auto_arima(gasTrain,seasonal=True,m=12,n_jobs=-1)
arima_model.summary()
Out[29]:
SARIMAX Results
Dep. Variable: y No. Observations: 358
Model: SARIMAX(1, 1, 1)x(1, 0, [], 12) Log Likelihood -957.346
Date: Thu, 10 Aug 2023 AIC 1922.691
Time: 22:37:36 BIC 1938.202
Sample: 01-01-1990 HQIC 1928.860
- 10-01-2019
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.4428 0.041 10.812 0.000 0.363 0.523
ma.L1 -0.8979 0.035 -25.549 0.000 -0.967 -0.829
ar.S.L12 -0.1163 0.033 -3.527 0.000 -0.181 -0.052
sigma2 12.4585 0.468 26.629 0.000 11.542 13.375
Ljung-Box (L1) (Q): 0.08 Jarque-Bera (JB): 1891.78
Prob(Q): 0.78 Prob(JB): 0.00
Heteroskedasticity (H): 2.19 Skew: 0.62
Prob(H) (two-sided): 0.00 Kurtosis: 14.21


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
In [30]:
arima_model.plot_diagnostics(figsize=(15,7))
plt.show()
In [31]:
# use the model to make predictions
pred_in_sample = arima_model.predict_in_sample()
pred_out_sample = arima_model.predict(n_periods=len(gasTest))
pred_full = np.concatenate((pred_in_sample, pred_out_sample))
# Plot the results
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(gasTrain, label='training')
ax.plot(gasTest, label='testing')
ax.plot(hDF.index[1:], pred_full[1:], label='prediction')
plt.legend()
plt.title('Auto ARIMA Model')
plt.show()
In [32]:
# To evaluate the auto arima model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(gasTrain, pred_in_sample)
mape_test = mean_absolute_percentage_error(gasTest, pred_out_sample)

print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')

Model Mean Absolute Percentage Error on training data is 11.19%
Model Mean Absolute Percentage Error on testing data is 15.39%

Gas Consumption Sarima model¶

Sarimax is a seasonal ARIMA model¶

In [33]:
# SARIMAX model
from statsmodels.tsa.statespace.sarimax import SARIMAX
sarimax_model = SARIMAX(gasTrain,order=(2,1,1),seasonal_order=(1,1,1,12)).fit()
sarimax_model.summary()

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
Out[33]:
SARIMAX Results
Dep. Variable: Gas Consumption (tons) No. Observations: 358
Model: SARIMAX(2, 1, 1)x(1, 1, 1, 12) Log Likelihood -948.040
Date: Thu, 10 Aug 2023 AIC 1908.079
Time: 22:37:56 BIC 1931.141
Sample: 01-01-1990 HQIC 1917.263
- 10-01-2019
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.4427 0.048 9.233 0.000 0.349 0.537
ar.L2 0.0314 0.058 0.542 0.587 -0.082 0.145
ma.L1 -0.9022 0.043 -21.140 0.000 -0.986 -0.819
ar.S.L12 -0.1049 0.042 -2.508 0.012 -0.187 -0.023
ma.S.L12 -0.9996 6.995 -0.143 0.886 -14.710 12.711
sigma2 12.5396 87.516 0.143 0.886 -158.988 184.067
Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 2508.97
Prob(Q): 0.99 Prob(JB): 0.00
Heteroskedasticity (H): 2.46 Skew: 0.96
Prob(H) (two-sided): 0.00 Kurtosis: 16.07


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
In [34]:
sarimax_model.plot_diagnostics(figsize=(15,7))
plt.show()
In [35]:
pred = sarimax_model.get_prediction(start='1990-01-01', end='2023-01-01').predicted_mean
pred = pred[1:]
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(gasTrain, label='training')
ax.plot(gasTest, label='testing')
ax.plot(pred, label='prediction')
plt.legend()
plt.title('SARIMA Model')
plt.show()
In [36]:
# To evaluate the auto arima model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(gasTrain[1:], pred[pred.index<'2019-11'])
mape_test = mean_absolute_percentage_error(gasTest, pred[pred.index>='2019-11'])
rmse_train = np.sqrt(mean_absolute_percentage_error(gasTrain[1:], pred[pred.index<'2019-11']))
rmse_test = np.sqrt(mean_absolute_percentage_error(gasTest, pred[pred.index>='2019-11']))
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')
print(f'RMSE on training data is {rmse_train:.2f}')
print(f'RMSE on testing data is {rmse_test:.2f}')

Model Mean Absolute Percentage Error on training data is 11.86%
Model Mean Absolute Percentage Error on testing data is 14.45%
RMSE on training data is 0.34
RMSE on testing data is 0.38

Sarimax Model With Log Transformation¶

In [37]:
# SARIMAX model
from statsmodels.tsa.statespace.sarimax import SARIMAX
sarimax_model = SARIMAX(np.log(gasTrain), order=(2,1,1),seasonal_order=(1,1,1,12)).fit()
sarimax_model.summary()

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
Out[37]:
SARIMAX Results
Dep. Variable: Gas Consumption (tons) No. Observations: 358
Model: SARIMAX(2, 1, 1)x(1, 1, 1, 12) Log Likelihood 118.895
Date: Thu, 10 Aug 2023 AIC -225.791
Time: 22:38:06 BIC -202.730
Sample: 01-01-1990 HQIC -216.607
- 10-01-2019
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.4322 0.049 8.900 0.000 0.337 0.527
ar.L2 0.0375 0.057 0.658 0.511 -0.074 0.149
ma.L1 -0.9088 0.037 -24.521 0.000 -0.981 -0.836
ar.S.L12 -0.0879 0.038 -2.288 0.022 -0.163 -0.013
ma.S.L12 -0.9939 0.450 -2.210 0.027 -1.875 -0.112
sigma2 0.0260 0.011 2.299 0.022 0.004 0.048
Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 539.63
Prob(Q): 0.98 Prob(JB): 0.00
Heteroskedasticity (H): 1.71 Skew: -0.05
Prob(H) (two-sided): 0.00 Kurtosis: 9.13


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

Predictions are scaled back for us to intepret the results.¶

In [38]:
pred = sarimax_model.get_prediction(start='1990-01-01', end='2023-01-01').predicted_mean
pred = np.exp(pred)
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(gasTrain, label='training')
ax.plot(gasTest, label='testing')
ax.plot(pred, label='prediction')
plt.legend()
plt.title('SARIMA Model')
plt.show()
In [39]:
# To evaluate the auto arima model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(gasTrain, pred[pred.index<'2019-11'])
mape_test = mean_absolute_percentage_error(gasTest, pred[pred.index>='2019-11'])
rmse_train = np.sqrt(mean_absolute_percentage_error(gasTrain, pred[pred.index<'2019-11']))
rmse_test = np.sqrt(mean_absolute_percentage_error(gasTest, pred[pred.index>='2019-11']))
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')
print(f'RMSE on training data is {rmse_train:.2f}')
print(f'RMSE on testing data is {rmse_test:.2f}')

Model Mean Absolute Percentage Error on training data is 13.10%
Model Mean Absolute Percentage Error on testing data is 16.05%
RMSE on training data is 0.36
RMSE on testing data is 0.40

Since the MAPE increased, we will not use log transformation.¶

Gas Consumption AR Model¶

In [40]:
from statsmodels.tsa.ar_model import AutoReg
arModel = AutoReg(gasTrain,lags=2)
arModelFit = arModel.fit()
print(arModelFit.summary())

                              AutoReg Model Results                               
==================================================================================
Dep. Variable:     Gas Consumption (tons)   No. Observations:                  358
Model:                         AutoReg(2)   Log Likelihood                -960.562
Method:                   Conditional MLE   S.D. of innovations              3.594
Date:                    Thu, 10 Aug 2023   AIC                           1929.124
Time:                            22:38:11   BIC                           1944.624
Sample:                        03-01-1990   HQIC                          1935.290
                             - 10-01-2019                                         
=============================================================================================
                                coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------------------
const                         6.5060      1.015      6.412      0.000       4.517       8.495
Gas Consumption (tons).L1     0.5521      0.053     10.509      0.000       0.449       0.655
Gas Consumption (tons).L2     0.1700      0.052      3.242      0.001       0.067       0.273
                                    Roots                                    
=============================================================================
                  Real          Imaginary           Modulus         Frequency
-----------------------------------------------------------------------------
AR.1            1.2948           +0.0000j            1.2948            0.0000
AR.2           -4.5419           +0.0000j            4.5419            0.5000
-----------------------------------------------------------------------------

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
In [41]:
pred = arModelFit.get_prediction(start='1990-01-01', end='2023-01-01').predicted_mean
pred = pred[2:] #Remove First 2 NaN Values
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(gasTrain, label='training')
ax.plot(gasTest, label='testing')
ax.plot(pred, label='prediction')
plt.legend()
plt.title('AR Model')
plt.show()
In [42]:
# To evaluate the AR model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(gasTrain[2:], pred[pred.index<'2019-11'])
mape_test = mean_absolute_percentage_error(gasTest, pred[pred.index>='2019-11'])
rmse_train = np.sqrt(mean_absolute_percentage_error(gasTrain[2:], pred[pred.index<'2019-11']))
rmse_test = np.sqrt(mean_absolute_percentage_error(gasTest, pred[pred.index>='2019-11']))
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')
print(f'RMSE on training data is {rmse_train:.2f}')
print(f'RMSE on testing data is {rmse_test:.2f}')

Model Mean Absolute Percentage Error on training data is 11.33%
Model Mean Absolute Percentage Error on testing data is 16.15%
RMSE on training data is 0.34
RMSE on testing data is 0.40

Since our SARIMA model has the lowest AIC and BIC as well as the lowest MAPE, we will further try to improve our model by finding the best order and seasonal order for it.¶

For Loop to find the Best Order for the Gas' SARIMAX model¶

In [168]:
p_values = range(1,4)
d_values = range(1,2)
q_values = range(1,4)
P_values = range(1,4)
D_values = range(1,2)
Q_values = range(1,4)
seasonality = [12]
bestMapeOrder = None
bestAICOrder= None
bestBICOrder = None
bestAICBICOrder = None
bestAIC = 100000000
bestMape = 10000000
bestBIC = 1000000
for p in p_values:
    for d in d_values:
        for q in q_values:
            for P in P_values:
                for D in D_values:
                    for Q in Q_values:
                        for s in seasonality:
                            order=(p,d,q)
                            seasonalOrder=(P,D,Q,s)
                        try:
                            
                            model = SARIMAX(gasTrain,order=order,seasonal_order=seasonalOrder).fit()
                            predictions = model.forecast(steps=len(gasTest))
                            mape = mean_absolute_percentage_error(gasTest, predictions)
                            
                            if(model.aic<bestAIC):
                                bestAIC = model.aic
                                bestAICOrder= f'({order},{seasonalOrder})'
                            if(model.bic<bestBIC):
                                bestBIC = model.bic
                                bestBICOrder= f'({order},{seasonalOrder})'
                            if(model.bic<bestBIC and model.aic<bestAIC):
                                bestAICBICOrder= f'({order},{seasonalOrder})'
                                
                            if(mape<bestMape):
                                bestMapeaic = model.aic
                                bestMape = mape
                                bestMapeOrder= f'({order},{seasonalOrder})'  
                            
                                
                        except Exception as e:
                            print(f'LU Decomposition Error: {order},{seasonalOrder}')
                            

print(f'Order for Best AIC:{bestAICOrder}')
print(f'Order for Best BIC:{bestBICOrder}')
print(f'Order for Best AIC and BIC:{bestAICBICOrder}')
print(f'Order for Best MAPE:{bestMapeOrder}')
                            
                            

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)

LU Decomposition Error: (1, 1, 2),(2, 1, 3, 12)

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "

Order for Best AIC:((1, 1, 1),(1, 1, 1, 12))
Order for Best BIC:((1, 1, 1),(1, 1, 1, 12))
Order for Best AIC and BIC:None
Order for Best MAPE:((3, 1, 1),(3, 1, 1, 12))

Gas' SARIMA Model (After Improvement)¶

In [43]:
# SARIMAX model
from statsmodels.tsa.statespace.sarimax import SARIMAX
sarimax_model = SARIMAX(gasTrain, order=(1,1,1),seasonal_order=(1,1,1,12)).fit()
sarimax_model.summary()

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
Out[43]:
SARIMAX Results
Dep. Variable: Gas Consumption (tons) No. Observations: 358
Model: SARIMAX(1, 1, 1)x(1, 1, 1, 12) Log Likelihood -948.172
Date: Thu, 10 Aug 2023 AIC 1906.345
Time: 22:38:49 BIC 1925.563
Sample: 01-01-1990 HQIC 1913.998
- 10-01-2019
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.4477 0.043 10.317 0.000 0.363 0.533
ma.L1 -0.8948 0.036 -25.041 0.000 -0.965 -0.825
ar.S.L12 -0.1115 0.042 -2.649 0.008 -0.194 -0.029
ma.S.L12 -0.9997 9.888 -0.101 0.919 -20.379 18.380
sigma2 12.5475 123.852 0.101 0.919 -230.197 255.292
Ljung-Box (L1) (Q): 0.06 Jarque-Bera (JB): 2482.43
Prob(Q): 0.81 Prob(JB): 0.00
Heteroskedasticity (H): 2.45 Skew: 0.94
Prob(H) (two-sided): 0.00 Kurtosis: 16.01


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
In [44]:
pred = sarimax_model.get_prediction(start='1990-01-01', end='2023-01-01').predicted_mean
pred = pred[1:]
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(gasTrain, label='training')
ax.plot(gasTest, label='testing')
ax.plot(pred, label='prediction')
plt.legend()
plt.title('SARIMA Model')
plt.show()
In [45]:
# To evaluate the auto arima model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(gasTrain[1:], pred[pred.index<'2019-11'])
mape_test = mean_absolute_percentage_error(gasTest, pred[pred.index>='2019-11'])
rmse_train = np.sqrt(mean_absolute_percentage_error(gasTrain[1:], pred[pred.index<'2019-11']))
rmse_test = np.sqrt(mean_absolute_percentage_error(gasTest, pred[pred.index>='2019-11']))
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')
print(f'RMSE on training data is {rmse_train:.2f}')
print(f'RMSE on testing data is {rmse_test:.2f}')

Model Mean Absolute Percentage Error on training data is 11.86%
Model Mean Absolute Percentage Error on testing data is 14.25%
RMSE on training data is 0.34
RMSE on testing data is 0.38

Water Consumption (tons)¶

Exponential Smoothing (As a Baseline)¶

In [46]:
expoModel = ExponentialSmoothing(waterTrain,trend='add',seasonal='add',seasonal_periods=12).fit()
pred = expoModel.forecast(len(waterTest))
mape = mean_absolute_percentage_error(waterTest,pred)*100

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
In [47]:
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(waterTrain.index,waterTrain, label='training')
ax.plot(waterTest.index,waterTest, label='testing')
ax.plot(waterTest.index, pred, label='prediction')
plt.legend()
plt.title(' Water Consumption\'s Exponential Smoothing')
plt.show()
In [48]:
print(f'Test Data Mean Absolute Percentage Error: {mape:.2f}%')

Test Data Mean Absolute Percentage Error: 18.47%

Water Consumption's ACF and PACF plot¶

In [49]:
seasonalWaterDF = df.copy()
seasonalWaterDF = seasonalWaterDF[['Water Consumption (tons)']]
seasonalWaterDF['Seasonal Diff'] = seasonalWaterDF['Water Consumption (tons)'].diff(12)
seasonalWaterDF = seasonalWaterDF.dropna()

#Seasonal acf and pacf
plot_acf(seasonalWaterDF['Seasonal Diff'],lags=50)
plot_pacf(seasonalWaterDF['Seasonal Diff'])
plt.show()

Tail off at ACF Plot, which indicates an AR model. From PACF plot, cut off happens at lag 2. This illustrates an AR(2) model¶

In [50]:
from pmdarima.arima import auto_arima
arima_model = auto_arima(waterTrain,seasonal=True,m=12,n_jobs=-1)
arima_model.summary()
Out[50]:
SARIMAX Results
Dep. Variable: y No. Observations: 358
Model: SARIMAX(1, 1, 2)x(2, 0, [], 12) Log Likelihood -2156.629
Date: Thu, 10 Aug 2023 AIC 4325.259
Time: 22:39:41 BIC 4348.525
Sample: 01-01-1990 HQIC 4334.513
- 10-01-2019
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.6593 0.110 5.997 0.000 0.444 0.875
ma.L1 -1.2279 0.120 -10.210 0.000 -1.464 -0.992
ma.L2 0.2631 0.101 2.598 0.009 0.065 0.462
ar.S.L12 0.0620 0.054 1.156 0.248 -0.043 0.167
ar.S.L24 0.1468 0.065 2.244 0.025 0.019 0.275
sigma2 1.03e+04 534.503 19.262 0.000 9248.063 1.13e+04
Ljung-Box (L1) (Q): 0.01 Jarque-Bera (JB): 110.57
Prob(Q): 0.93 Prob(JB): 0.00
Heteroskedasticity (H): 1.69 Skew: -0.36
Prob(H) (two-sided): 0.00 Kurtosis: 5.63


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
In [51]:
arima_model.plot_diagnostics(figsize=(15,7))
plt.show()
In [52]:
# use the model to make predictions
pred_in_sample = arima_model.predict_in_sample()
pred_out_sample = arima_model.predict(n_periods=len(waterTest))
pred_full = np.concatenate((pred_in_sample, pred_out_sample))
pred_full = pred_full[1:] 
# Plot the results
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(waterTrain, label='training')
ax.plot(waterTest, label='testing')
ax.plot(hDF.index[1:], pred_full, label='prediction')
plt.legend()
plt.title('Auto ARIMA Model')
plt.show()
In [53]:
# To evaluate the auto arima model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(waterTrain, pred_in_sample)
mape_test = mean_absolute_percentage_error(waterTest, pred_out_sample)
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')

Model Mean Absolute Percentage Error on training data is 21.53%
Model Mean Absolute Percentage Error on testing data is 17.50%

Water Exogenous Variables¶

Exogenous variables are external factors that might indirectly affect water's time series¶

In [54]:
waterExogenousTrain = gasTrain
waterExogenousTest = gasTest
In [55]:
from statsmodels.tsa.statespace.sarimax import SARIMAX
sarimax_model = SARIMAX(waterTrain,exog=waterExogenousTrain, order=(1,1,2),seasonal_order=(2,1,1,12)).fit()
sarimax_model.summary()

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
Out[55]:
SARIMAX Results
Dep. Variable: Water Consumption (tons) No. Observations: 358
Model: SARIMAX(1, 1, 2)x(2, 1, [1], 12) Log Likelihood -2080.181
Date: Thu, 10 Aug 2023 AIC 4176.362
Time: 22:39:53 BIC 4207.111
Sample: 01-01-1990 HQIC 4188.608
- 10-01-2019
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
Gas Consumption (tons) -11.0173 0.962 -11.458 0.000 -12.902 -9.133
ar.L1 0.7407 0.060 12.282 0.000 0.623 0.859
ma.L1 -1.2433 0.121 -10.276 0.000 -1.480 -1.006
ma.L2 0.2446 0.087 2.821 0.005 0.075 0.415
ar.S.L12 0.1129 0.070 1.618 0.106 -0.024 0.250
ar.S.L24 0.1210 0.063 1.914 0.056 -0.003 0.245
ma.S.L12 -0.9502 0.069 -13.849 0.000 -1.085 -0.816
sigma2 9291.5697 953.835 9.741 0.000 7422.088 1.12e+04
Ljung-Box (L1) (Q): 0.04 Jarque-Bera (JB): 33.70
Prob(Q): 0.85 Prob(JB): 0.00
Heteroskedasticity (H): 1.73 Skew: -0.34
Prob(H) (two-sided): 0.00 Kurtosis: 4.37


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
In [56]:
pred = sarimax_model.get_prediction(start='1990-01-01', end='2023-01-01',exog = waterExogenousTest).predicted_mean
pred = pred[1:]
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(waterTrain, label='training')
ax.plot(waterTest, label='testing')
ax.plot(pred, label='prediction')
plt.legend()
plt.title('SARIMA Model')
plt.show()
In [57]:
sarimax_model.plot_diagnostics(figsize=(15,7))
plt.show()
In [58]:
# To evaluate the auto arima model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(waterTrain[1:], pred[pred.index<'2019-11'])
mape_test = mean_absolute_percentage_error(waterTest, pred[pred.index>='2019-11'])
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')

Model Mean Absolute Percentage Error on training data is 19.53%
Model Mean Absolute Percentage Error on testing data is 15.52%

AR(2) Model¶

In [59]:
from statsmodels.tsa.ar_model import AutoReg
arModel = AutoReg(waterTrain,lags=2)
arModelFit = arModel.fit()
print(arModelFit.summary())

                               AutoReg Model Results                                
====================================================================================
Dep. Variable:     Water Consumption (tons)   No. Observations:                  358
Model:                           AutoReg(2)   Log Likelihood               -2160.946
Method:                     Conditional MLE   S.D. of innovations            104.704
Date:                      Thu, 10 Aug 2023   AIC                           4329.892
Time:                              22:39:58   BIC                           4345.392
Sample:                          03-01-1990   HQIC                          4336.058
                               - 10-01-2019                                         
===============================================================================================
                                  coef    std err          z      P>|z|      [0.025      0.975]
-----------------------------------------------------------------------------------------------
const                         146.4027     22.474      6.514      0.000     102.354     190.452
Water Consumption (tons).L1     0.4979      0.052      9.578      0.000       0.396       0.600
Water Consumption (tons).L2     0.1974      0.052      3.797      0.000       0.096       0.299
                                    Roots                                    
=============================================================================
                  Real          Imaginary           Modulus         Frequency
-----------------------------------------------------------------------------
AR.1            1.3189           +0.0000j            1.3189            0.0000
AR.2           -3.8411           +0.0000j            3.8411            0.5000
-----------------------------------------------------------------------------

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
In [60]:
pred = arModelFit.get_prediction(start='1990-01-01', end='2023-01-01').predicted_mean
pred = pred[2:] #Remove First 2 NaN Values
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(waterTrain, label='training')
ax.plot(waterTest, label='testing')
ax.plot(pred, label='prediction')
plt.legend()
plt.title('AR Model')
plt.show()
In [61]:
# To evaluate the AR model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(waterTrain[2:], pred[pred.index<'2019-11'])
mape_test = mean_absolute_percentage_error(waterTest, pred[pred.index>='2019-11'])
rmse_train = np.sqrt(mean_absolute_percentage_error(waterTrain[2:], pred[pred.index<'2019-11']))
rmse_test = np.sqrt(mean_absolute_percentage_error(waterTest, pred[pred.index>='2019-11']))
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')
print(f'RMSE on training data is {rmse_train:.2f}')
print(f'RMSE on testing data is {rmse_test:.2f}')

Model Mean Absolute Percentage Error on training data is 22.05%
Model Mean Absolute Percentage Error on testing data is 18.32%
RMSE on training data is 0.47
RMSE on testing data is 0.43

For Loop to find the Best Order for the Water's SARIMAX model¶

In [62]:
p_values = range(1,4)
d_values = range(1,2)
q_values = range(1,4)
P_values = range(1,4)
D_values = range(1,2)
Q_values = range(1,4)
seasonality = [12]
bestMapeOrder = None
bestAICOrder= None
bestBICOrder = None
bestAIC = 100000000
bestMape = 10000000
bestBIC = 1000000
for p in p_values:
    for d in d_values:
        for q in q_values:
            for P in P_values:
                for D in D_values:
                    for Q in Q_values:
                        for s in seasonality:
                            order=(p,d,q)
                            seasonalOrder=(P,D,Q,s)
                        try:
                            
                            model = SARIMAX(waterTrain,exog=waterExogenousTrain,order=order,seasonal_order=seasonalOrder).fit()
                            predictions = model.forecast(steps=len(waterTest),exog=waterExogenousTest)
                            mape = mean_absolute_percentage_error(waterTest, predictions)
                            
                            if(model.aic<bestAIC):
                                bestAIC = model.aic
                                bestAICOrder= f'({order},{seasonalOrder})'
                            if(model.bic<bestBIC):
                                bestBIC = model.bic
                                bestBICOrder= f'({order},{seasonalOrder})'
                                
                            if(mape<bestMape):
                                bestMapeaic = model.aic
                                bestMape = mape
                                bestMapeOrder= f'({order},{seasonalOrder})'  
                            
                                
                        except Exception as e:
                            print(f'LU Decomposition Error: {order},{seasonalOrder}')
                            

print(f'Order for Best AIC:{bestAICOrder}')
print(f'Order for Best BIC:{bestBICOrder}')
print(f'Order for Best MAPE:{bestMapeOrder}')
                            
                            

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "

Order for Best AIC:((1, 1, 1),(2, 1, 1, 12))
Order for Best BIC:((1, 1, 1),(2, 1, 1, 12))
Order for Best MAPE:((2, 1, 3),(2, 1, 2, 12))

We will choose order (2,1,3),(2,1,2,12) for best MAPE, although the AIC and BIC will be abit higher, it will be a good trade off for better accuracy.¶

Water's SARIMA Model (After Improvement)¶

In [63]:
from statsmodels.tsa.statespace.sarimax import SARIMAX
sarimax_model = SARIMAX(waterTrain,exog=waterExogenousTrain, order=(2,1,3),seasonal_order=(2,1,2,12)).fit()
sarimax_model.summary()

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
Out[63]:
SARIMAX Results
Dep. Variable: Water Consumption (tons) No. Observations: 358
Model: SARIMAX(2, 1, 3)x(2, 1, [1, 2], 12) Log Likelihood -2080.853
Date: Thu, 10 Aug 2023 AIC 4183.705
Time: 22:49:18 BIC 4225.984
Sample: 01-01-1990 HQIC 4200.543
- 10-01-2019
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
Gas Consumption (tons) -10.2715 0.980 -10.483 0.000 -12.192 -8.351
ar.L1 -0.2424 0.059 -4.137 0.000 -0.357 -0.128
ar.L2 0.7568 0.059 12.935 0.000 0.642 0.871
ma.L1 -0.2822 1.749 -0.161 0.872 -3.711 3.146
ma.L2 -0.9840 1.241 -0.793 0.428 -3.417 1.449
ma.L3 0.2663 0.485 0.550 0.583 -0.683 1.216
ar.S.L12 -0.4140 0.417 -0.993 0.320 -1.231 0.403
ar.S.L24 0.1578 0.067 2.350 0.019 0.026 0.289
ma.S.L12 -0.4431 0.428 -1.036 0.300 -1.282 0.395
ma.S.L24 -0.4987 0.378 -1.318 0.187 -1.240 0.243
sigma2 9165.6061 1.58e+04 0.579 0.562 -2.18e+04 4.02e+04
Ljung-Box (L1) (Q): 0.11 Jarque-Bera (JB): 28.54
Prob(Q): 0.74 Prob(JB): 0.00
Heteroskedasticity (H): 1.72 Skew: -0.31
Prob(H) (two-sided): 0.00 Kurtosis: 4.27


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
In [64]:
pred = sarimax_model.get_prediction(start='1990-01-01', end='2023-01-01',exog = waterExogenousTest).predicted_mean
pred = pred[1:]
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(waterTrain, label='training')
ax.plot(waterTest, label='testing')
ax.plot(pred, label='prediction')
plt.legend()
plt.title('SARIMA Model')
plt.show()
In [65]:
# To evaluate the SARIMA model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(waterTrain[1:], pred[pred.index<'2019-11'])
mape_test = mean_absolute_percentage_error(waterTest, pred[pred.index>='2019-11'])
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')

Model Mean Absolute Percentage Error on training data is 19.63%
Model Mean Absolute Percentage Error on testing data is 15.15%

Electricity Consumption (MWh)¶

Exponential Smoothing (As a Baseline)¶

In [66]:
from statsmodels.tsa.holtwinters import ExponentialSmoothing
from sklearn.metrics import mean_absolute_percentage_error
expoModel = ExponentialSmoothing(electricityTrain,trend='multiplicative',seasonal='multiplicative',seasonal_periods=12).fit()
pred = expoModel.forecast(len(electricityTest))
mape = mean_absolute_percentage_error(electricityTest,pred)*100

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
In [67]:
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(electricityTrain.index,electricityTrain, label='training')
ax.plot(electricityTest.index,electricityTest, label='testing')
ax.plot(electricityTest.index, pred, label='prediction')
plt.legend()
plt.title('Electricity Consumption\'s Exponential Smoothing')
plt.show()
In [68]:
print(f'Test Data Mean Absolute Percentage Error: {mape:.2f}%')

Test Data Mean Absolute Percentage Error: 3.99%

Electricity's ACF and PACF Plot¶

Electricity is first differenced then plugged into the ACF and PACF Plot.¶

In [69]:
seasonalElectricityDF = df.copy()
seasonalElectricityDF = seasonalElectricityDF[['Electricity Consumption (MWh)']]
seasonalElectricityDF['Seasonal Diff'] = seasonalElectricityDF['Electricity Consumption (MWh)'].diff(12)
seasonalElectricityDF = seasonalElectricityDF.dropna()

#Seasonal acf and pacf
plot_acf(seasonalElectricityDF['Seasonal Diff'],lags=50)
plot_pacf(seasonalElectricityDF['Seasonal Diff'])
plt.show()

Tail off at ACF Plot, which indicates an AR model. From PACF plot, cut off happens at lag 1. This illustrates an AR(1) model¶

Electricity Auto Arima¶

In [70]:
from pmdarima.arima import auto_arima
arima_model = auto_arima(electricityTrain,seasonal=True,m=12,n_jobs=-1)
arima_model.summary()
Out[70]:
SARIMAX Results
Dep. Variable: y No. Observations: 358
Model: SARIMAX(2, 1, 1)x(1, 0, 1, 12) Log Likelihood -1630.415
Date: Thu, 10 Aug 2023 AIC 3274.831
Time: 22:50:21 BIC 3301.975
Sample: 01-01-1990 HQIC 3285.627
- 10-01-2019
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
intercept 0.0030 0.009 0.356 0.722 -0.014 0.020
ar.L1 0.5783 0.056 10.293 0.000 0.468 0.688
ar.L2 -0.1063 0.055 -1.933 0.053 -0.214 0.002
ma.L1 -0.9503 0.019 -51.105 0.000 -0.987 -0.914
ar.S.L12 0.9945 0.005 219.480 0.000 0.986 1.003
ma.S.L12 -0.7362 0.044 -16.894 0.000 -0.822 -0.651
sigma2 496.5159 30.662 16.193 0.000 436.419 556.613
Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 27.29
Prob(Q): 0.98 Prob(JB): 0.00
Heteroskedasticity (H): 2.78 Skew: -0.11
Prob(H) (two-sided): 0.00 Kurtosis: 4.34


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
In [71]:
arima_model.plot_diagnostics(figsize=(15,7))
plt.show()
In [72]:
# use the model to make predictions
pred_in_sample = arima_model.predict_in_sample()
pred_out_sample = arima_model.predict(n_periods=len(electricityTest))
pred_full = np.concatenate((pred_in_sample, pred_out_sample))
# Plot the results
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(electricityTrain, label='training')
ax.plot(electricityTest, label='testing')
ax.plot(hDF.index[1:], pred_full[1:], label='prediction')
plt.legend()
plt.title('Auto ARIMA Model')
plt.show()
In [73]:
# To evaluate the auto arima model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(electricityTrain, pred_in_sample)
mape_test = mean_absolute_percentage_error(electricityTest, pred_out_sample)
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')

Model Mean Absolute Percentage Error on training data is 2.36%
Model Mean Absolute Percentage Error on testing data is 2.95%

Electricity's SARIMA Model¶

In [74]:
# # SARIMAX model
from statsmodels.tsa.statespace.sarimax import SARIMAX
sarimax_model = SARIMAX(electricityTrain,order=(2,1,1),seasonal_order=(2,1,2,12)).fit()
sarimax_model.summary()

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
Out[74]:
SARIMAX Results
Dep. Variable: Electricity Consumption (MWh) No. Observations: 358
Model: SARIMAX(2, 1, 1)x(2, 1, [1, 2], 12) Log Likelihood -1558.475
Date: Thu, 10 Aug 2023 AIC 3132.950
Time: 22:50:38 BIC 3163.698
Sample: 01-01-1990 HQIC 3145.196
- 10-01-2019
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.5474 0.057 9.683 0.000 0.437 0.658
ar.L2 -0.0573 0.055 -1.039 0.299 -0.165 0.051
ma.L1 -0.9453 0.021 -44.512 0.000 -0.987 -0.904
ar.S.L12 0.4796 0.212 2.259 0.024 0.064 0.896
ar.S.L24 -0.2809 0.067 -4.191 0.000 -0.412 -0.150
ma.S.L12 -1.1796 0.220 -5.373 0.000 -1.610 -0.749
ma.S.L24 0.4333 0.175 2.471 0.013 0.090 0.777
sigma2 472.4280 31.156 15.163 0.000 411.362 533.494
Ljung-Box (L1) (Q): 0.01 Jarque-Bera (JB): 24.77
Prob(Q): 0.94 Prob(JB): 0.00
Heteroskedasticity (H): 2.66 Skew: -0.18
Prob(H) (two-sided): 0.00 Kurtosis: 4.26


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
In [75]:
sarimax_model.plot_diagnostics(figsize=(15,7))
plt.show()
In [76]:
pred = sarimax_model.get_prediction(start='1990-01-01', end='2023-01-01').predicted_mean
pred = pred[1:]
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(electricityTrain, label='training')
ax.plot(electricityTest, label='testing')
ax.plot(pred, label='prediction')
plt.legend()
plt.title('SARIMA Model')
plt.show()
In [77]:
# To evaluate the auto arima model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(electricityTrain[1:], pred[pred.index<'2019-11'])
mape_test = mean_absolute_percentage_error(electricityTest, pred[pred.index>='2019-11'])
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')

Model Mean Absolute Percentage Error on training data is 2.16%
Model Mean Absolute Percentage Error on testing data is 2.40%

Electricity's AR(1) Model¶

In [78]:
from statsmodels.tsa.ar_model import AutoReg
arModel = AutoReg(electricityTrain,lags=1)
arModelFit = arModel.fit()
print(arModelFit.summary())

                                  AutoReg Model Results                                  
=========================================================================================
Dep. Variable:     Electricity Consumption (MWh)   No. Observations:                  358
Model:                                AutoReg(1)   Log Likelihood               -2035.032
Method:                          Conditional MLE   S.D. of innovations             72.344
Date:                           Thu, 10 Aug 2023   AIC                           4076.064
Time:                                   22:50:46   BIC                           4087.697
Sample:                               02-01-1990   HQIC                          4080.691
                                    - 10-01-2019                                         
====================================================================================================
                                       coef    std err          z      P>|z|      [0.025      0.975]
----------------------------------------------------------------------------------------------------
const                              109.5366     22.561      4.855      0.000      65.318     153.755
Electricity Consumption (MWh).L1     0.8751      0.025     34.341      0.000       0.825       0.925
                                    Roots                                    
=============================================================================
                  Real          Imaginary           Modulus         Frequency
-----------------------------------------------------------------------------
AR.1            1.1427           +0.0000j            1.1427            0.0000
-----------------------------------------------------------------------------

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
In [79]:
pred = arModelFit.get_prediction(start='1990-01-01', end='2023-01-01').predicted_mean
pred = pred[3:] #Remove First 3 NaN Values
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(electricityTrain, label='training')
ax.plot(electricityTest, label='testing')
ax.plot(pred, label='prediction')
plt.legend()
plt.title('AR Model')
plt.show()
In [80]:
# To evaluate the AR model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(electricityTrain[3:], pred[pred.index<'2019-11'])
mape_test = mean_absolute_percentage_error(electricityTest, pred[pred.index>='2019-11'])
rmse_train = np.sqrt(mean_absolute_percentage_error(electricityTrain[3:], pred[pred.index<'2019-11']))
rmse_test = np.sqrt(mean_absolute_percentage_error(electricityTest, pred[pred.index>='2019-11']))
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')
print(f'RMSE on training data is {rmse_train:.2f}')
print(f'RMSE on testing data is {rmse_test:.2f}')

Model Mean Absolute Percentage Error on training data is 6.98%
Model Mean Absolute Percentage Error on testing data is 13.56%
RMSE on training data is 0.26
RMSE on testing data is 0.37

Since SARIMA Model have the lowest AIC, BIC and MAPE, we will tune the SARIMAX Model to find its best order and seasonal order.¶

For Loop To Find The Best Order For Electricity's Sarimax Model¶

In [151]:
p_values = range(1,4)
d_values = range(1,2)
q_values = range(1,4)
P_values = range(1,4)
D_values = range(1,2)
Q_values = range(1,4)
seasonality = [12]
bestMapeOrder = None
bestAICOrder= None
bestBICOrder = None
bestAIC = 100000000
bestMape = 10000000
bestBIC = 1000000
for p in p_values:
    for d in d_values:
        for q in q_values:
            for P in P_values:
                for D in D_values:
                    for Q in Q_values:
                        for s in seasonality:
                            order=(p,d,q)
                            seasonalOrder=(P,D,Q,s)
                        try:
                            
                            model = SARIMAX(electricityTrain,order=order,seasonal_order=seasonalOrder).fit()
                            predictions = model.forecast(steps=len(electricityTest))
                            mape = mean_absolute_percentage_error(electricityTest, predictions)
                            
                            if(model.aic<bestAIC):
                                bestAIC = model.aic
                                bestAICOrder= f'({order},{seasonalOrder})'
                            if(model.bic<bestBIC):
                                bestBIC = model.bic
                                bestBICOrder= f'({order},{seasonalOrder})'
                            if(mape<bestMape):
                                bestMapeaic = model.aic
                                bestMape = mape
                                bestMapeOrder= f'({order},{seasonalOrder})'  
                            
                                
                        except Exception as e:
                            print(f'LU Decomposition Error: {order},{seasonalOrder}')
                            

print(f'Order for Best AIC:{bestAICOrder}')
print(f'Order for Best BIC:{bestBICOrder}')
print(f'Order for Best MAPE:{bestMapeOrder}')
                            
                            

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "

Order for Best AIC:((3, 1, 3),(2, 1, 2, 12))
Order for Best BIC:((1, 1, 1),(2, 1, 1, 12))
Order for Best MAPE:((3, 1, 2),(3, 1, 2, 12))

We will choose order (3,1,3),(2,1,2,12) for our SARIMAX which has the best AIC.¶

Electricity Sarimax Model (After Improvement)¶

In [81]:
# SARIMAX model
from statsmodels.tsa.statespace.sarimax import SARIMAX
sarimax_model = SARIMAX(electricityTrain,order=(3,1,3),seasonal_order=(2,1,2,12)).fit()
sarimax_model.summary()

C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
C:\Users\kieny\anaconda3\lib\site-packages\statsmodels\base\model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "
Out[81]:
SARIMAX Results
Dep. Variable: Electricity Consumption (MWh) No. Observations: 358
Model: SARIMAX(3, 1, 3)x(2, 1, [1, 2], 12) Log Likelihood -1554.616
Date: Thu, 10 Aug 2023 AIC 3131.233
Time: 22:51:03 BIC 3173.512
Sample: 01-01-1990 HQIC 3148.070
- 10-01-2019
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 -1.1834 0.198 -5.992 0.000 -1.571 -0.796
ar.L2 0.1169 0.129 0.909 0.364 -0.135 0.369
ar.L3 0.3193 0.123 2.593 0.010 0.078 0.561
ma.L1 0.8059 0.174 4.624 0.000 0.464 1.148
ma.L2 -0.8720 0.043 -20.231 0.000 -0.956 -0.787
ma.L3 -0.7240 0.159 -4.542 0.000 -1.036 -0.412
ar.S.L12 0.5498 0.175 3.143 0.002 0.207 0.893
ar.S.L24 -0.2803 0.065 -4.282 0.000 -0.409 -0.152
ma.S.L12 -1.2989 0.182 -7.122 0.000 -1.656 -0.941
ma.S.L24 0.5286 0.147 3.605 0.000 0.241 0.816
sigma2 465.0290 30.781 15.108 0.000 404.699 525.359
Ljung-Box (L1) (Q): 0.02 Jarque-Bera (JB): 31.00
Prob(Q): 0.88 Prob(JB): 0.00
Heteroskedasticity (H): 2.52 Skew: -0.29
Prob(H) (two-sided): 0.00 Kurtosis: 4.35


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
In [82]:
pred = sarimax_model.get_prediction(start='1990-01-01', end='2023-01-01').predicted_mean
pred = pred[1:]
fig, ax = plt.subplots(figsize=(14, 5))
ax.plot(electricityTrain, label='training')
ax.plot(electricityTest, label='testing')
ax.plot(pred, label='prediction')
plt.legend()
plt.title('SARIMA Model')
plt.show()
In [83]:
# To evaluate the auto arima model
from sklearn.metrics import mean_absolute_percentage_error
mape_train = mean_absolute_percentage_error(electricityTrain[1:], pred[pred.index<'2019-11'])
mape_test = mean_absolute_percentage_error(electricityTest, pred[pred.index>='2019-11'])
print(f'Model Mean Absolute Percentage Error on training data is {mape_train*100:.2f}%')
print(f'Model Mean Absolute Percentage Error on testing data is {mape_test*100:.2f}%')

Model Mean Absolute Percentage Error on training data is 2.14%
Model Mean Absolute Percentage Error on testing data is 2.31%

Conclusion¶

Final Model Chosen:

Gas Consumption: SARIMAX(1,1,1)x(1,1,1,12)

Water Consumption: SARIMAX(2,1,3)x(2,1,2,12)

Electricity Usage: SARIMAX(3,1,3)x(2,1,2,12)

My electricity consumption forecast is the most accurate with the lowest MAPE score of 2.14% on training data and 2.31% on testing data. Compared to the exponential smoothing that was used as a baseline, my MAPE has decreased significantly which implies that my model is more accurate when forecasting.¶

All in all, this time series project has allowed me to learn more in-depth about forecasting a time series and how forecasting is important for decision-makers to make the best decision.¶

In [ ]:
In [ ]: